Blow-up phenomena and peakons for the b-family of FORQ/MCH equations

Danchin, 2005, Fourier Analysis Methods for PDEs Bahouri, 2011, Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343 Kato, 1975, Quasi-linear equations of evolution, with applications to partial differential equations, vol. 448, 25 Holm, 2003, Nonlinear balance and exchange of stability of dynamics of solitons, peakons, ramps/cliffs and leftons in a 1+1 nonlinear evolutionary PDE, Phys. Lett. A, 308, 437, 10.1016/S0375-9601(03)00114-2 Holm, 2003, Wave structure and nonlinear balance in a family of 1+1 evolutionary PDE's, SIAM J. Appl. Dyn. Syst., 2, 323, 10.1137/S1111111102410943 Dullin, 2004, On asymptotically equivalent shallow water wave equations, Phys. D: Nonlinear Phenom., 190, 1, 10.1016/j.physd.2003.11.004 Fokas, 1981, Symplectic structures, their Bäklund transformation and hereditary symmetries, Phys. D, 4, 47, 10.1016/0167-2789(81)90004-X Camassa, 1993, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71, 1661, 10.1103/PhysRevLett.71.1661 Fisher, 1999, The Camassa–Holm equation: conserved quantities and the initial value problem, Phys. Lett. A, 259, 371, 10.1016/S0375-9601(99)00466-1 Dai, 1998, Model equations for nolinear dispersive waves in a compressible Mooney–Rivlin rod, Acta Mech., 127, 193, 10.1007/BF01170373 Constantin, 2001, On the scattering problem for the Camassa–Holm equation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 457, 953, 10.1098/rspa.2000.0701 Qiao, 2003, The Camassa–Holm hierarchy, N-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold, Comm. Math. Phys., 239, 309, 10.1007/s00220-003-0880-y Constantin, 2006, Inverse scattering transform for the Camassa–Holm equation, Inverse Probl., 22, 2197, 10.1088/0266-5611/22/6/017 Monvel, 2009, Long-time asymptotics for the Camassa–Holm equation, SIAM J. Math. Anal., 41, 1559, 10.1137/090748500 Alber, 1994, The geometry of peaked solitons and billiard solutions of a class of integrable PDE's, Lett. Math. Phys., 32, 137, 10.1007/BF00739423 Cao, 2004, Traveling wave solutions for a class of one-dimensional nonlinear shallow water wave models, J. Dynam. Differential Equations, 16, 167, 10.1023/B:JODY.0000041284.26400.d0 Constantin, 2006, The trajectories of particles in Stokes waves, Invent. Math., 166, 23, 10.1007/s00222-006-0002-5 Qiao, 2006, On peaked and smooth solitons for the Camassa–Holm equation, Europhys. Lett., 73, 657, 10.1209/epl/i2005-10453-y Constantin, 2007, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44, 423, 10.1090/S0273-0979-07-01159-7 Constantin, 2011, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173, 559, 10.4007/annals.2011.173.1.12 Kouranbaeva, 1998, The Camassa–Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys., 40, 857, 10.1063/1.532690 Misiolek, 1998, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24, 203, 10.1016/S0393-0440(97)00010-7 Constantin, 2002, Stability of the Camassa–Holm solitons, J. Nonlinear Sci., 12, 415, 10.1007/s00332-002-0517-x Constantin, 2000, Stability of peakons, Comm. Pure Appl. Math., 53, 603, 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L Bressan, 2007, Global conservative solutions of the Camassa–Holm equation, Arch. Ration. Mech. Anal., 183, 215, 10.1007/s00205-006-0010-z Bressan, 2007, Global dissipative solutions of the Camassa–Holm equation, Anal. Appl., 5, 1, 10.1142/S0219530507000857 Li, 2000, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162, 27, 10.1006/jdeq.1999.3683 Danchin, 2001, A few remarks on the Camassa–Holm equation, Differential Integral Equations, 14, 953, 10.57262/die/1356123175 Danchin, 2003, A note on well-posedness for Camassa–Holm equation, J. Differential Equations, 192, 429, 10.1016/S0022-0396(03)00096-2 Constantin, 1998, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181, 229, 10.1007/BF02392586 Constantin, 1998, Global existence and blow-up for a shallow water equation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 26, 303 Constantin, 2000, Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier, 50, 321, 10.5802/aif.1757 Constantin, 2000, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z., 233, 75, 10.1007/PL00004793 Degasperis, 1998, Asymptotic integrability, 23 Degasperis, 2002, A new integrable equation with peakon solutions, Theoret. Math. Phys., 133, 1463, 10.1023/A:1021186408422 Qiao, 2004, Integrable hierarchy (the DP hierarchy), 3×3 constrained systems, and parametric and stationary solutions, Acta Appl. Math., 83, 199, 10.1023/B:ACAP.0000038872.88367.dd Qiao, 2004, A new integrable hierarchy (short wave model for the DP), parametric solution, and traveling wave solution, Math. Phys. Anal. Geom., 7, 289, 10.1007/s11040-004-3090-8 Constantin, 2009, The hydrodynamical relevant of the Camassa–Holm and Degasperis–Procesi equations, Arch. Ration. Mech. Anal., 192, 165, 10.1007/s00205-008-0128-2 Lundmark, 2005, Multi-peakon solutions of the Degasperis–Procesi equation, Inverse Probl., 19, 1241, 10.1088/0266-5611/19/6/001 Lenells, 2005, Traveling wave solutions of the Degasperis–Procesi equation, J. Math. Anal. Appl., 306, 72, 10.1016/j.jmaa.2004.11.038 Zhang, 2007, Cuspons and smooth solitons of the Degasperis–Procesi equation under inhomogeneous boundary condition, Math. Phys. Anal. Geom., 10, 205, 10.1007/s11040-007-9027-2 Lin, 2010, Stability of peakons for the Degasperis–Procesi equation, Comm. Pure Appl. Math., 62, 125 Zhou, 2004, Blow-up phenomenon for the integrable Degasperis–Procesi equation, Phys. Lett. A, 328, 157, 10.1016/j.physleta.2004.06.027 Escher, 2006, Global weak solutions and blow-up structure for the Degasperis–Procesi equation, J. Funct. Anal., 241, 457, 10.1016/j.jfa.2006.03.022 Liu, 2006, Global existence and blow-up phenomena for the Degasperis–Procesi equation, Comm. Math. Phys., 267, 801, 10.1007/s00220-006-0082-5 Qiao, 2006, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys., 47, 112701, 10.1063/1.2365758 Qiao, 2007, New integrable hierarchy, its parametric solutions, cuspons, one-peak solitons, and M/W-shape peak solitons, J. Math. Phys., 48, 10.1063/1.2759830 Fokas, 1995, On a class of physically important integrable equations, Phys. D, 87, 145, 10.1016/0167-2789(95)00133-O Fuchssteiner, 1996, Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa–Holm equation, Phys. D, 95, 229, 10.1016/0167-2789(96)00048-6 Olver, 1996, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53, 1900, 10.1103/PhysRevE.53.1900 Gui, 2013, Wave-breaking and peakons for a modified Camassa–Holm equation, Comm. Math. Phys., 319, 731, 10.1007/s00220-012-1566-0 Fu, 2013, On the Cauchy problem for the integrable modified Camassa–Holm equation with cubic nonlinearity, J. Differential Equations, 255, 1905, 10.1016/j.jde.2013.05.024 Himonas, 2014, The Cauchy problem for the Fokas–Olver–Rosenau–Qiao equation, Nonlinear Anal., 95, 499, 10.1016/j.na.2013.09.028 Zhang, 2016, Global wellposedness of cubic Camassa–Holm equations, Nonlinear Anal., 133, 61, 10.1016/j.na.2015.11.020 Himonas, 2014, Höder continuity for the Fokas–Olver–Rosenau–Qiao equation, J. Nonlinear Sci., 24, 1105, 10.1007/s00332-014-9212-y Chen, 2015, Oscillation-induced blow-up to the modified Camassa–Holm equation with linear dispersion, Adv. Math., 272, 225, 10.1016/j.aim.2014.12.003